An update on the coin-moving game on the square grid

TL;DR

This paper advances the understanding of coin-moving puzzles on square grids by correcting previous theorems, introducing new solution methods, and analyzing complex cases with one or two extra coins.

Contribution

It revises earlier results on puzzles with two extra coins, presents a new solving method, and explores complex puzzles with one extra coin on the square grid.

Findings

Counterexamples to previous theorems on two extra coins

A new method for solving certain two-extra-coin puzzles

Analysis of specific one-extra-coin puzzles

Abstract

This paper extends the work started in 2002 by Demaine, Demaine and Verill (DDV) on coin-moving puzzles. These puzzles have a long history in the recreational literature, but were first systematically analyzed by DDV, who gave a full characterization of the solvable puzzles on the triangular grid and a partial characterization of the solvable puzzles on the square grid. This article specifically extends the study of the game on the square grid. Notably, DDV's result on puzzles with two "extra coins" is shown to be overly broad: this paper provides counterexamples as well as a revised version of this theorem. A new method for solving puzzles with two extra coins is then presented, which covers some cases where the aforementioned theorem does not apply. Puzzles with just one extra coin seem even more complicated, and are only touched upon by DDV. This paper delves deeper, studying a class of such puzzles that may be considered equivalent to a game of "poking" coins. Within this class, some cases are considered that are amenable to analysis.